Which of the following is a possible length for side AB of triangle AB

The third side of a triangle should be less than the sum of the other 2 sides and greater than the mod(difference of 2 sides)

so, x should lie between mod(y-z)<x<y+z
(9-6)<x<9+6
3<x<15

only value is 13.5

2 and 3 for me as well .... we need the third side to be between 3 and 15, i.e. 3<x<15

Both 2 and 3 fit this requirement

9(3^1/2) = 15.58>15 ...only 3.

yes, how silly of me

Rules for a triangle
(1)Third side is always greater than the difference of two other sides
(2) Third side is always less than the sum of two other sides
Going by above rules only 13.5 satisfies the condition.

The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.

(9 - 6) < AB < (9 + 6)
3 < AB < 15.

Only 13.5 is in this range (\(9\sqrt{3}\approx{9*1.7}=15.3>15\)).

Answer: C.

Its still not convincing for me to consider C as the correct answer because no where the question states AB is the third side or the longest side.. If for example we consider the AB = 3 (answer Choice A), The side of the triangle would be 3,6 and 9.. Now 9 becomes the third side and satisfies all the constraints of the third side.. The same is applicable for the rest two choices.. I feel answer in E.. Can someone explain whats the assumption am missing here ?

Hi ragunandan,

You have to be very careful with the logic here.

If you have sides of 3, 6 and 9, then you DO NOT actually have a triangle - you have a line segment with a length of 9 "on top of" another line segment with a length of 9. You can use a ruler and physically draw it on a piece of paper if that will help you to visualize the issue.

From a "math" standpoint, if 9 is the longest side, then the other two sides have to SUM to a value that is GREATER than 9 (otherwise you don't have a triangle) and a DIFFERENCE that is LESS than 9.

GMAT assassins aren't born, they're made,
Rich

Hi Rich,
Thanks for your nice explanations you provide but I did not understand 'you have a line segment with a length of 9 "on top of" another line segment with a length of 9. You can use a ruler and physically draw it on a piece of paper if that will help you to visualize the issue.' I tried to draw it but it is just incomplete triangle. is that the result you mean?

Also I want to know if you cover those tactics and tricks for geometry in your courses.

Thanks in advance for your help

Hi ragunandan,

Which of the following is a possible length for side AB of triangle ABC if AC = 6 and BC = 9?

1. It's mentioned in the question statement that AB is the side of triangle ABC. Since we have been given two sides AC and BC, AB has to the third side.

2. Secondly since the question asks the possible length of side AB, we need to consider all possible values of AB i.e. AB can be the shortest side or the longest side of triangle ABC.

3. Lastly AB = 3 is not a legible value of the 3rd side of triangle ABC with the other two sides as AC = 6 and BC = 9. In a triangle the sum of two sides should be greater than the 3rd side. In this case AB + AC = 9 = BC. Similarly AB = 15√3 is also not a legible value. Among the given options AB = 13.5 is the only value for which we can form a triangle with the other two sides being 6 and 9.

Let me know in case you still have any trouble in the explanation

Regards
Harsh

Hi Mo2men,

You've discovered another variation on 'problem' of having sides of 3, 6 and 9 - if you tried to form a triangle, then the 3 and the 6 would NOT touch, so you would NOT have a triangle.

This all stems from a math rule called the Triangle Inequality Theorem; it's relatively rare on the GMAT (you might not even see it on Test Day; if you do see it, you'll probably see it just once). Thus, the concept is not a big point-gainer on the GMAT, so unless you're already scoring at a really high level in the Quant, you should be focused on other subjects.

To answer your last question: yes, we do cover it (and a couple of variants of it) in the EMPOWERgmat Course.

GMAT assassins aren't born, they're made,
Rich

Hi Rich, Thanks for the detailed explanation. #Helpful

Hi Harsh,Thanks for the explanation.. It was simpler and neat..

POINT : The effort here is to make the rule of triangle more precise and less time consuming

We know the rule in different forms however we only need to know the rule in just one line

i.e. SUM OF TWO SHORTER SIDES > LONGEST SIDE OF THE TRIANGLE

FOR THE TRIANGLE TO EXIST

Let's apply this in this case

For Finding the range of values of the Third side

For minimum value of the third side AB, We must consider the given BC = 9 as the longest side

Hence AB (Min) + AC > BC

i.e. AB (Min) + 6 > 9

i.e. AB (Min) > 3

For Maximum value of the third side AB, We must consider the given BC = 9 and Ac=6 as the two shorter sides

Hence AC + BC > AB (Max.)

i.e. 6 + 9 > AB (Max.)

i.e. AB (Max.) < 15

i.e. the range of AB becomes

3 < AB < 15

I. 3 NOT ACCEPTABLE
II. 9 √3 - 9x1.73 = 15.3 NOT ACCEPTABLE
III. 13.5 ACCEPTABLE

Answer: Option

The third side of a triangle must be less than the sum of the other two sides and greater than their difference (i.e. |y - z| < x < y + z).
In this question:
|BC - AC| < AB < BC + AC
9 - 6 < AB < 9 + 6
3 < AB < 15
Only 13.5 is in this range. 9sqrt3 is approximately equal to 9(1.7) or 15.3.
The correct answer is C.

We can approach this problem with triangle properties in mind; though, first it is important to note that the stem does not state that this triangle is a right triangle so the Pythagorean Theorem cannot be applied:

(I) The sums of the two smaller sides of a triangle must be larger than the hypotenuse or greatest side- "3" fails this test and is therefore not a viable option
(II) If considered 9 √3 as 9(1.7) = 15.3 which is larger than 6 + 9 and thus not a viable option
(III) 13.5 is smaller than the sum of 9 and 6 and therefore a viable option

The correct answer is (C)

As pointed out earlier, the question is based on the triangle inequality theorem:
In that case AB has to be smaller than the sum of the two given sides 9+6=15 and be bigger than their difference 9-6=3
Thus you can write the inequality as follows: 3<AB<15.
If you now test the options only Roman Numeral III fits in that range, so the answer is C.